AI system autonomously discovers new insights on computing limits
At the Limits of Computability: Steps Towards Autonomous AI Discovery on the Busy Beaver Frontier
Artificial IntelligenceFormal Languages and Automata Theory
Summary
Some problems in math and computer science are so complex that no computer program can always solve them. The authors built an AI system that can explore these hard problems on its own, finding new rules and verifying them mathematically. Their AI worked on special computational machines called Busy Beavers, discovering new patterns that we didn’t know before. This shows how AI could help find new math ideas without human prompts.
What this means in practice
- •For software developers: Create tools that automatically find and verify complex patterns in code execution to help optimize algorithms.
- •For computational mathematicians: Use autonomous AI to explore open-ended problems and generate new conjectures with formal proofs.
Authors
Marco Dos Santos, Vivek Veeriah, Alex Havrilla, Andrew Bolt, Goran Zuzic, Satinder Singh, Tom Zahavy
Abstract
Recent breakthroughs in AI for mathematics---ranging from Olympiad medal performances to resolving long-standing open conjectures---have rapidly expanded what automated systems can achieve. Yet these successes primarily concern predefined, human-formulated problems, leaving open how AI systems can autonomously uncover mathematical structure and formulate new questions. To push beyond static challenges, we explore the Busy Beaver (BB) domain, an uncomputable frontier that is fundamentally open-ended. Because no algorithm can compute the Busy Beaver number $\mathrm{BB}(n)$ for every $n$, this domain provides a natural setting for sustained exploration through mathematical conjecture and proof. In this paper, we introduce a Gemini-based agentic pipeline that automatically discovers accelerations of Turing machines, expressed as compact mathematical rules summarizing many execution steps. Our system integrates pattern recognition, automated conjecturing, informal mathematical reasoning, and formal theorem proving in Lean. With this framework, our system autonomously discovered and formally verified novel Collatz-like accelerations for eleven machines in $\mathrm{BB}(6)$. Subject to community acceptance, we provisionally classify these machines as Cryptids, namely machines whose blank-tape executions admit exact, compact mathematical descriptions but whose halting behavior remains unresolved. These results advance the analysis of $\mathrm{BB}(6)$ holdouts on the Busy Beaver frontier and demonstrate the potential of autonomous AI systems for open-ended mathematical discovery in fundamental computer science.